1. (Indefinite) Integration
“Add Math” Calculus
(Indefinite) Integration

2. We first need to consider an example of differentiation
e.g.1 Differentiate (a) (b)
(a)
Equal !
(b)
The gradient functions are the same since the graph of is a just a translation of

3. Graphs of the functions
e.g. the gradient at
x = -1 is -2
At each value of x, the gradients of the 2 graphs are the same

4. C is called the arbitrary constant
Indefinite integration is the reverse of differentiation
If we are given the gradient function and want to find the equation of the curve, we reverse the process of differentiation
BUT the constant is unknown
So,
C is called the arbitrary constant
or constant of integration
The equation forms a family of curves

5. e.g.2 Find the equation of the family of curves which have a gradient function given by
Solution:
To reverse the rule of differentiation:
add 1 to the power
divide by the new power

6. Tip: Check the answer by differentiating
e.g.2 Find the equation of the family of curves which have a gradient function given by
Solution:
To reverse the rule of differentiation:
add 1 to the power
divide by the new power
add C
Tip: Check the answer by differentiating

7. The graphs look like this:
The gradient function
( Sample of 6 values of C )

8. We can only find the value of C if we have some additional information
e.g. 3 Find the equation of the family of curves with gradient function
Solution: The index of x in the term 3x is 1, so adding 1 to the index gives 2.
The constant -1 has no x. It integrates to -x.
We can only find the value of C if we have some additional information

9. Exercises
Find the equations of the family of curves with the following gradient functions: 1. 2. 3.
N.B. Multiply out the brackets first

10. Exercises
Find the equations of the family of curves with the following gradient functions: 1. 2. 3.

11. Finding the value of C
e.g.1 Find the equation of the curve which passes through the point (1, 2) and has gradient function given by
Solution:

12. 6 is the common denominator
Finding the value of C
e.g.1 Find the equation of the curve which passes through the point (1, 2) and has gradient function given by
Solution:
(1, 2) is on the curve:
6 is the common denominator

13. Exercises 1.
Find the equation of the curve with gradient function
which passes through the point ( 2, -2 )
Find the equation of the curve with gradient function
which passes through the point ( 2, 1 ) 2.

14. Solutions 1.
Ans:
( 2, -2 ) lies on the curve

15. Solutions 2.
( 2, 1 ) on the curve
So,

16. Notation for Integration
e.g. 1 We know that
Another way of writing integration is:
Called the integral sign
We read this as “d x ”.
It must be included to indicate that the variable is x
In full, we say we are integrating
“ with respect to x “.

17. e.g. 2 Find (a)
(b)
Solution:
( Integrate with respect to x )
(a)
(b)
( Integrate with respect to t )
e.g. 3 Integrate with respect to x
Solution:
The notation for integration must be written
We have done the integration so there is no integral sign

18. Exercises
1. Find
(a)
(b)
2. Integrate the following with respect to x:
(a)
(b)

19. Summary
Indefinite Integration is the reverse of differentiation.
A constant of integration, C, is always included.
Indefinite Integration is used to find a family of curves.
To find the curve through a given point, the value of C is found by substituting for x and y.
There are 2 notations:

20. Summary
Indefinite Integration is the reverse of differentiation.
A constant of integration, C, is always included.
There are 2 notations: